For most places where the distinction matters: A professional photographer is someone who makes a major or substantial part of this income from photography.

You see this definition often in rules for photo contest, professional association or for qualifying for professional service programs. For example:

• Photo contests me be open to amateurs only, professionals only or have separate competitions for amateurs and professionals.
• Professional associations of photographers often stipulate this in membership rules to make sure of the level of its members and focus events and discussions better.
• Camera manufacturers have professional services which entitle professionals to faster service, replacement loaners, etc.

The main consideration to answer your question is who is asking and why? You can certainly be a professional without high-end equipment and you can be a professional without formal training. After all, there is no mandates photography certification program.

For someone who is hiring you, it may simply be that you have experience and a portfolio that shows you are capable of the quality of images they seek.